Subgroups — #26127891

Order 12032.

246 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245187	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20241458	C₂	1	2	yes	maximal, complemented
4	#20241452	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
10	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
12	C₄×C₄₇⋊C₁₆	—	2	1	no	—
13	C₄₇×#431	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#322	C₄	1	2	yes	—
17	C₈²×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	1	no	—
21	C₂×C₄₇⋊C₁₆	—	4	2	no	—
22	C₂×M₁₆×C₄₇	C₈	1	1	yes	characteristic
23	C₂×M₁₆×C₄₇	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	1	no	—
28	C₂²⋊C₈×C₄₇	—	2	1	no	—
29	#97×C₄₇	C₈	1	2	yes	—
30	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
39	M₁₆×C₄₇	—	2	2	no	—
40	M₁₆×C₄₇	C₁₆	1	4	yes	complemented
41	C₂×C₈×C₄₇	—	2	1	no	—
42	C₂×C₈×C₄₇	—	2	1	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	4	1	no	—
45	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
54	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₄₇⋊C₁₆	—	4	1	no	—
56	C₄₇⋊C₁₆	—	4	2	no	—
57	C₄₇⋊C₁₆	—	4	1	no	—
58	C₂³×C₄₇	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	4	1	no	—
62	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
67	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₄₇	—	2	1	no	—
71	C₈×C₄₇	—	2	1	no	—
72	C₈×C₄₇	—	2	1	no	—
73	C₈×C₄₇	—	2	2	no	—
74	C₈×C₄₇	—	4	2	no	—
75	C₈×C₄₇	C₂×C₁₆	1	4	yes	complemented
76	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₈×C₄₇	#137	1	1	yes	characteristic
78	#12015	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	1	no	—
80	C₂²×C₄₇	—	2	1	no	—
81	C₂²×C₄₇	C₄⋊C₁₆	1	2	yes	—
82	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	—	2	1	no	—
86	C₄×C₄₇	C₄⋊C₁₆	1	2	yes	—
87	C₄×C₄₇	#428	1	1	yes	characteristic
88	C₄×C₄₇	#430	1	1	yes	characteristic
89	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
90	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
91	#2162	—	47	1	no	—
92	C₂×C₄⋊C₁₆	—	47	1	no	—
93	C₈×M₁₆	D₄₇	1	1	yes	characteristic
94	#1370	—	47	2	no	—
95	#1367	—	47	2	no	—
96	C₂×C₄₇	—	2	2	no	—
97	C₂×C₄₇	#1565	1	1	yes	characteristic
98	C₂×C₄₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
99	C₂×C₄₇	#2153	1	1	yes	characteristic
100	C₂²×C₁₆	—	94	1	no	—
101	#431	Dic₄₇	1	1	yes	characteristic
102	C₄×M₁₆	Dic₄₇	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
104	C₄⋊C₁₆	—	47	1	no	—
105	C₄⋊C₁₆	—	47	2	no	—
106	C₄⋊C₁₆	—	47	1	no	—
107	C₄⋊C₁₆	—	94	1	no	—
108	C₂²⋊C₁₆	—	94	1	no	—
109	C₄×C₁₆	—	94	1	no	—
110	#322	Dic₄₇	1	2	yes	—
111	C₈²	C₂×D₄₇	1	2	yes	—
112	C₄₇	#12015	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	C₄₇⋊C₈	1	1	yes	characteristic
114	C₂×M₁₆	C₄₇⋊C₈	1	1	yes	characteristic
115	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
116	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
118	C₂×C₁₆	—	94	1	no	—
119	C₂×C₁₆	—	94	1	no	—
120	C₂×C₁₆	—	94	2	no	—
121	C₂×C₁₆	—	188	2	no	—
122	C₄⋊C₈	—	2	1	no	—
123	C₂²⋊C₈	—	2	1	no	—
124	#97	C₄₇⋊C₈	1	2	yes	—
125	C₄×C₈	Dic₉₄	1	1	yes	characteristic
126	C₄×C₈	D₁₈₈	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
128	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
129	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
130	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
132	C₂²×C₄	#10625329	1	1	yes	characteristic
133	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	C₄₇⋊C₁₆	1	4	yes	complemented
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
141	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
142	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
145	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
146	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
147	C₄²	#10625329	1	1	yes	characteristic
148	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
149	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
150	C₁₆	—	188	1	no	—
151	C₁₆	—	188	2	no	—
152	C₁₆	—	188	1	no	—
153	C₂³	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
158	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
159	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
161	C₂×C₄	#13093964	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
164	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	2	no	—
170	C₈	C₂×C₄₇⋊C₁₆	1	4	yes	complemented
171	C₈	#13094052	1	1	yes	characteristic
172	C₈	#13094057	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊(C₄₇⋊C₁₆)	1	2	yes	—
176	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₄⋊(C₄₇⋊C₁₆)	1	2	yes	—
181	C₄	#15689165	1	1	yes	characteristic
182	C₄	#15689172	1	1	yes	characteristic
183	C₄	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
184	C₄	C₂×#13093964	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#20242140	1	1	yes	characteristic
187	C₂	C₂×C₄⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
188	C₂	#20245154	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₈
prime radical	C₂³×C₄₇	C₄⋊C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#12015
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20241458	C₂	semidirect
4. #20241452	C₂	semidirect
40. M₁₆×C₄₇	C₁₆	semidirect
75. C₈×C₄₇	C₂×C₁₆	semidirect
112. C₄₇	#12015	semidirect
135. M₁₆	C₄₇⋊C₁₆	semidirect
170. C₈	C₂×C₄₇⋊C₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20245187	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20241458	2
#20241452	2
C₈×M₁₆×C₄₇	2
#12015	47